I am getting the confusion with the definition of algebra. When we say $A$ is a finitely generated $R$- algebra then is that mean $A$ has a ring structure and finitely generated as an $R$-module. Thanks

1 Answer
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No, being finitely generated as an algebra is generally not as strong as being finitely generated as a module.

Being finitely generated as an algebra means that there is some finite set of elements from the algebra, such that the subalgebra generated by those elements is the entire algebra.

This means that apart from $R$-linear combinations of the elements, we can also take all products of the elements, which may well give us a lot more elements.

As an easy example of an algebra (let's say over a field $k$) that is finitely generated as an algebra but not finitely generated as a module over $k$, we can take the polynomial ring $k[x]$ in one indeterminate. As a $k$-module (ie, a vector space), this is infinite dimensional, so not finitely generated. But as a $k$-algebra, it is generated by the element $x$.

$\begingroup$If I remember right, those who use "finitely generated algebra" this way like to use "module-finite algebra" for the OP's meaning.$\endgroup$
– rschwiebJan 16 '13 at 18:37

$\begingroup$@Tobias: $A$ is an algebra over a field $F$. For finitely many elements $a_1,a_2,\cdots,a_n$ in the algebra, the sub-algebra generated by $a_1,a_2,\cdots,a_n$ we mean, we take $F$-linear combinations of them as well as their products, am I right?$\endgroup$
– GroupsOct 26 '15 at 9:41

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$\begingroup$@Groups Yes, all possible products and $F$-linear combinations (more precisely, we take the smallest subalgebra containing those elements, which will have a slightly different description if we leave out some of the "usual" requirements on the algebra).$\endgroup$
– Tobias KildetoftOct 26 '15 at 9:43